Do I need a permit to add a 10' x 10' shed to my backyard in Connecticut?

It will depend on several factors.
-If you live in the city limits check with your local building authority. Usually these are found at city hall or with a local utility company. Typically, if you do not live in the city limits there will not be building restrictions.
-If you live in a subdivision with a HOA make sure to check the guild lines about out buildings and permanent structures. Even an HOA outside of city limits may have building guild lines.
-If neither of these apply you may check with the county office just to make sure.

Less than 120 sq ft is not required to have a permit according to international building code.